Journal of Economic Methodology ISSN 1350178X print/ISSN 14699427 online © 2000 Taylor & Francis Ltd
http://www/tandf.co.uk/journals
Journal of Economic Methodology 7:2, 195–210 2000
Three attitudes towards data mining
Kevin D. Hoover and Stephen J. Perez
Abstract
‘Data mining’ refers to a broad class of activities that have in common,
a search over different ways to process or package data statistically or econo
metrically with the purpose of making the final presentation meet certain design
criteria. We characterize three attitudes toward data mining: first, that it is to
be avoided and, if it is engaged in, that statistical inferences must be adjusted to
account for it; second, that it is inevitable and that the only results of any interest
are those that transcend the variety of alternative data mined specifications ( a view
associated with Leamer’s extremebounds analysis); and third, that it is essential
and that the only hope we have of using econometrics to uncover true economic
relationships is to be found in the intelligent mining of data. The first approach
confuses considerations of sampling distribution and considerations of epistemic
warrant and, reaches an unnecessarily hostile attitude toward data mining. The
second approach relies on a notion of robustness that has little relationship to truth:
there is no good reason to expect a true specification to be robust alternative
specifications. Robustness is not, in general, a carrier of epistemic warrant. The
third approach is operationalized in the generaltospecific search methodology
of the LSE school of econometrics. Its success demonstrates that intelligent data
mining is an important element in empirical investigation in economics.
Keywords:
data mining, extremebounds analysis, specification search, general
tospecific, LSE econometrics
1 INTRODUCTION
To practice data mining is to sin against the norms of econometrics, of
that there can be little doubt. That few have attempted to justify professional
abhorrence to data mining signifies nothing, few have felt any pressing need
to justify our abhorrence of theft either. What is for practical purposes beyond
doubt needs no special justification; and we learn that data mining is bad
econometric practice, just as we learn that theft is bad social practice, at
our mothers’ knees as it were. Econometric norms, like social norms, are
internalized in an environment in which explicit prohibitions, implicit
example and, many subtle pressures to conformity mold our morés. Models
of ‘good’ econometric practice, stray remarks in textbooks or lectures, stern
warnings from supervisors and referees, all teach us that datamining is
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abhorrent. All agree that theft is wrong, yet people steal and, they mine data.
So, from time to time moralists, political philosophers and legal scholars find
it necessary to raise the prohibition against theft out of its position as a back
ground presupposition of social life, to scrutinize its ethical basis, to dis
criminate among its varieties, to categorize various practices as falling inside
or outside the strictures that proscribe it. Similarly, the practice of data mining
has itself been scrutinized only infrequently (e.g., Leamer 1978; 1983, Mayer
1980, 1993; Lovell 1983; Hoover 1995). In this paper, we wish to characterize
the practice of data mining and three attitudes towards it. The first attitude is
the one that that we believe is the most common in the profession namely, data
mining is to be avoided and, if it is engaged in, we must adjust our statistical
inferences to account for it. The second attitude is that data mining is
inevitable and that the only results of any interest are those that transcend the
variety of alternative data mined specifications. The third attitude is that data
mining is essential and that the only hope that we have of using econometrics
to uncover true economic relationships is to be found in the intelligent mining
of data.
2 WHAT IS DATA MINING?
‘Data mining’ refers to a broad class of activities that have in common a search
over different ways to process or package data statistically or econometrically
with the purpose of making the final presentation meet certain design criteria.
An econometrician might try different combinations of regressors, different
sample periods, different functional forms, or different estimation methods
in order to find a regression that suited a theoretical preconception, had
‘significant’ coefficients, maximized goodnessoffit, or some other criterion
or set of criteria. To clarify the issues, consider a particularly common sort
of data mining exercise. The object of the search is the process that generates
y
, where
y
= [y
t
], an N
´
1 vector of observations, t = 1, 2, ... N. Let
X
= {X
j
},
j = 1, 2, ... M, be the universe of variables over which a search might be
conducted. Let
X
P
=
XX
, the power set of
X
(i.e., the set of all subsets of
X
).
If
y
were generated from a linear process, then the actual set of variables that
generated it is an element of
X
P
. Call this set of true determinants
X
T
Î
X
P
,
and let the true datagenerating process be:
y
k
=
X
k
T
b
T
+ v
k
,(1)
where
w
k
= [
w
k
t
], the vector of error terms, and k indicates the different
realizations of both errors and the variables in
X
. Now, let
X
i
Î
X
P
be any set
of variables; these define a model:
y
k
=
X
k
i
b
i
+ «
k
i
,(2)
where «
k
= [
e
k
t
] includes
w
k
, as well as every factor by which equation
(2) deviates from the true underlying process in equation (1). Typically, in
Three attitudes towards data mining 197
¯
economics only one realization of these variables is observed, k is degenerate
and takes only a single value. In other fields, for example in randomized
experiments in agriculture and elsewhere, k truly ranges over multiple realiz
ations, each realization it is assumed, coming from the same underlying
distribution. While in general, regressors might be random (the possibility
indicated by the superscript k on
X
i
), many analytical conclusions require the
assumption that
X
i
remains fixed in repeated samples of the error term.
1
This
amounts to,
X
k
i
=
X
h
i
,
;
k, h, while,
e
k
¹
e
h
,
;
k
¹
h, except on a set of measure
zero.
We can estimate the model in equation (2) for a given i and any particular
realization of the errors (a given k). From such estimations we can obtain
various sample statistics. For concreteness, consider the estimated standard
errors that correspond to b
ˆ
i
the estimated coefficients of equation (2) for
specification i.
2
What we would like to have are the population standard errors
of the elements of b
ˆ
i
about b
ˆ
i
. Conceptually, they are the dispersion of the
sampling distribution of the estimated coefficients while
X
i
remains fixed
in repeated samples of the error term. Ideally, sample distributions would
be calculated over a range of k’s, and as k approached infinity, the sample
distributions would converge to the population distributions. In practice there
is a single realization of
e
k
. While conceptually this requires a further
assumption that the errors at different times are drawn from the same distri
bution (the ergodic property), the correct counterfactual question remains:
what would the distribution be if it were possible to obtain multiple realiz
ations with fixed regressors? Conceptually, the distribution of sample statistics
is derived from repeatedly resampling the residual within a constant specifi
cation. This is clear in the case of standard errors estimated in Monte Carlo
settings or from bootstrap procedures.
3
In each case, simulations are pro
grammed that exactly mimic the analysis just laid out.
Datamining in this context amounts to searching over the various
X
i
Î
X
P
in order to meet selection criteria: e.g., that all of the tstatistics on the ele
ments of
X
i
be statistically significant or that R
2
be maximized.
3 ONLY OUR PREJUDICES SURVIVE
Data mining is considered reprehensible largely because the world is full of
accidental correlations, so that what a search turns up is thought to be more
a reflection of what we want to find than what is true about the world. A
methodology that emphasizes choice among a wide array of variables based
on their correlations is bound to select variables that just happen to be related
in the particular data set to the dependent variables, even though there is no
economic basis for the relationship. One response to this problem is to ban
search altogether. Econometrics is regarded as hypothesis testing. Only a well
specified model should be estimated and if it fails to support the hypothesis, it
fails; and the economist should not search for a better specification.
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A common variant of this view, however, recognizes that search is likely
and might even be useful or fruitful. However, it questions the meaning of the
test statistics associated with the final reported model. The implicit argument
runs something like this: Conventional test statistics are based on independent
draws. The tests in a sequence of tests on the same data used to guide the
specification search are not necessarily independent. The test statistics for
any specification that has survived such a process are necessarily going to be
‘significant’. They are ‘Darwinian’ in the sense that only the fittest survive.
Since we know in advance that they pass the tests, the critical values for the
tests could not possibly be correct. The critical values for such Darwinian test
statistics must in fact be much higher. The hard part is to quantify the appro
priate adjustment to the test statistics.
The interesting thing about this attitude towards data mining is the role that
it assigns to the statistics. In the presentation of the textbook interpretation
in the last section, those statistics were clearly reflections of sampling
distribution. Here the statistics are proposed as measures of epistemic
warrant, that is, as measures of our justification for believing a particular
specification to be the truth or as measures of the nearness of a particular
specification to the truth.
4
Sampling distribution is independent of the
investigator: it is a relationship between the particular specification and the
random errors thrown up by the world; the provenance of the specification
does not matter. Epistemic warrant is not independent of the investigator. To
take an extreme example, if we know an economist to be a prejudiced advocate
of a particular result and he presents us with a specification that confirms his
prejudice, our best guess is that the specification reflects the decision rule 
search until you find a confirming specification.
Not all search represents pure prejudice but, if test statistics are conceived
of as measures of epistemic warrant, the standard statistics will not be
appropriate in the presence of search. Michael Lovell (1983) provides an
example of this epistemic approach to test statistics. He argues that critical
values must be adjusted to reflect the degree of search. Lovell conducts a
number of simulations to make his point. In the first set of simulations, Lovell
(1983: 2
–
4) considers a regression like equation (2) in which the elements of
X
are mutually orthogonal. He considers sets of regressors that include exactly
two members (i.e., a fairly narrow subset of
X
P
). The dependent variable
y
is actually purely random and unrelated to any of the variables in
X
(i.e.,
X
T
=
Æ)
. Using five per cent critical values, he demonstrates that one or
more significant tstatistics occur more than five percent of the time. He pro
poses a formula to correct the critical values to account for the amount of
search.
Lovell (1983: 4
–
11) also considers simulations in which there are genuine
underlying relationships and the data are not mutually orthogonal. He uses a
data set of twenty actual macroeconomic series as the universe of search
X
.
Subsets of
X
(i.e.,
X
T
for the particular simulation)
with at most two members
Three attitudes towards data mining 199
are used to generate a simulated dependent variable. He then evaluates the
success of different search algorithms in recovering the particular variables
used to generate the dependent variable. These algorithms are different methods
for choosing a ‘best’ set of regressors as
X
i
ranges over the elements of
X
P
?
Success can be judged by the ability of an algorithm to recover
X
T
. Lovell
also tracks the coefficients on individual variables X
g
Î
X
, noting whether or
not they are statistically significant at conventional levels. He is, therefore,
able to report empirical type I and type II error rates (i.e., size and power). As
in his first simulation, he finds that there are substantial size distortions, so that
conventional critical values would be grossly misleading. What is more, he
finds low empirical power, which is related to the algorithms inability to
recover
X
T
. The critical point for our purposes is that Lovell’s simulations
implicitly interpret test statistics as measures of epistemic warrant. The standard
critical value or the size of the test refers to the probability of a particular
tstatistic on repeated draws of
w
k
(k taking on multiple values) from the same
distribution (that is the significance of the textbook assumption that the
regressors are fixed in repeated samples). Lovell’s experiment, in contrast,
takes the error term in the true datagenerating process,
w
k
, to be fixed (there
is a single k for each simulation) but, considers the way in which the distri
bution of
e
k
i
, the estimated residual for each specification considered in the
search process, varies with every new
X
i
. Lovell’s numbers are correct but the
question they answer refers to a particular application of a particular search
procedure rather than to any property of the specification independently in
relation to the world.
The difficulty with interpreting test statistics in this manner is that the
actual numbers are specific for a particular search procedure in a particular
context. This is obvious if we think about how Lovell or anyone would con
duct a Monte Carlo simulation to establish the modified critical values or sizes
of tests. A particular choice must be made for which variables appear in
X
and
a particular choice procedure must be adopted for searching over elements of
X
P
. Furthermore, one must establish a measure of the amount of search and
keep track of it. Yet, typically economists do not know how much search
produced any particular specification, nor is the universe of potential
regressors well defined. We do not start with a blank slate. Suppose, for
example, we estimate a ‘Goldfeld’ specification for money demand (Goldfeld
1973; also Judd and Scadding 1982). How many times has it been estimated
before? What do we know in advance of estimating it about how it is likely to
perform? What is the range of alternative specifications that have been or
might be considered? A specification such as the Goldfeld money demand
equation has involved literally incalculable amounts of search. Where would
we begin to assign epistemically relevant numbers to such a specification?
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4 ONLY THE ROBUST SHOULD SURVIVE
Edward Leamer (1978, 1983; and in Hendry et al. 1990) embraces the impli
cation of this last question. He suggests immersing empirical investigation in
the vulgarities of data mining in order to exploit the ability of a researcher
to produce differing estimates of coefficient values through repeated search.
Only if it is not possible for a researcher to eliminate an empirical finding
should it be believed. Leamer is a Bayesian. Yet, Bayesian econometrics
present a number of technical hurdles that prevent even many of those who,
like Leamer, believe that it is the correct way to proceed in principle from
applying it in practice. Instead, Leamer suggests a practicable alternative to
Bayesian statistics: extreme bounds analysis. The Bayesian question is, how
much incremental information is there in a set of data with which we might
update our beliefs? Leamer (1983) and Leamer and Leonard (1983) argue
that, if econometric conclusions are sensitive to alternative specifications,
then they do not carry much information useful for updating our beliefs. Data
may be divided into free variables, which theory suggests should be in a
regression; focus variables, a subset of the free variables which are of
immediate interest; and doubtful variables, which competing theories suggest
might be important.
5
Leamer suggests estimating specifications that corre
spond to every linear combination of doubtful variables in combination with
all of the free variables (including the focus variables). The extreme bounds of
the effects of the focus variables are given by the endpoints of the range of
values (
±
2 standard deviations) assigned to the coefficients on each of them
across these alternative regressions. If the extreme bounds are close together
then there can be some consensus on the import of the data for the problem at
hand; and if the extreme bounds are wide, that import is not pinned down very
precisely. If the extreme bounds bracket zero, then the direction of the effect is
not even clear. Such a variable can be regarded as not robust to alternative
specification.
6
The linkage between extreme bounds analysis and Bayesian principles
is not, however, onetoone in the sense that the central idea, robustness to
alternative specification, represents an attitude to datamining held by non
Bayesians as well. Thomas Mayer’s (1993, 2000) argument that every
regression run by an investigator, not just the final preferred specification,
ought to be reported arises from a similar notion of robustness. If a coefficient
is little changed under a variety of specifications, we should have confidence
in it, and not otherwise. Mayer’s proposal that the evidence ought not to
be suppressed, but reported, at least in a summary fashion (e.g., as extreme
bounds) is, he argues, an issue of honest communication and not a deep episte
mological problem. But we believe that this is incorrect. The epistemological
issue is this: if all the regressions are reported, just what is anyone supposed to
conclude from them?
The notion of robustness here is an odd one, as can be seen from a simple
Three attitudes towards data mining 201
example. Let A, B, and C be mutually orthogonal variables. Let a linear com
bination of the three and a random error term determine a fourth variable D.
Now if the coefficient on C relative to its variance is small compared with the
coefficients on A and B relative to their variances and, the variance of C is
small relative to the variance of the error term, then the coefficient on C may
have a low conventionally calculated tstatistic and a high standard error. C
has a low signaltonoise ratio. Let us suppose that C is just significant at a
conventional level of significance (say, five per cent) when the true specifi
cation is estimated. How will C fare under extreme bounds analysis? The
omission of A, B or both, is likely to raise the standard error substantially
and the point estimate of the coefficient on C plus or minus twice its standard
deviation might now bracket zero.
7
We would then conclude that C is not
a robust variable and that it is not possible to reach a consensus, even though
ex hypothesi it is a true determinant of D.
One response might be that it is just an unfortunate fact that sometimes
the data are not sufficiently discriminating. The lack of robustness of variable
C tells us that, while there may be a truth, we just do not have enough
information to narrow the range of prior beliefs about that truth, despite the
willingness of investigators to consider the complete range of possibilities.
The difference between the real world and the example here is that, unlike here
we never know the actual truth. Thus, if we happen to estimate the truth, yet
the truth is not robust, our true estimate carries little conviction or epistemic
warrant.
A second response, however, is that the example here illustrates that there is
no good reason to expect a true specification to be robust
–
that is, to be robust
to misspecification. Robustness is not, in general, a carrier of epistemic warrant.
Leamer (in Hendry et al. 1990: 188) attacks the very notion of a true specifi
cation:
I ... don’t think there is a true datagenerating process ...
To me the essential difference between the Bayesian and a classical point
of view is not that the parameters are treated as random variables, but
rather that the sampling distributions are treated as subjective distributions
or characterizations of states of mind ... And by ‘states of mind’ what I
mean is the opinion that it is useful for me to operate as if the data were
generated in a certain way.
Econometrics for Leamer is about characterizing the data but not about dis
covering the actual processes that generated the data. We find this position to
be barely coherent. The relationships among data are interesting only when
they go beyond the particular factual context in which they are estimated.
If we estimate a relationship between prices and quantities, for instance, we
might wish to use it predictively (what is our best estimate of tomorrow’s
price?) or counterfactually (if the price had been different, how would the
quantity have been different?). Either way, the relationship is meant to go
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beyond the observed data and apply with some degree of generality to an
unobserved domain. To say that there is a true datagenerating process is to
say that a specification could in principle, at least approximately, capture that
implied general relationship. To deny this would appear to defeat the purpose
of doing empirical economics. The very idea of a specification in which
different observations are connected by a common description seems to imply
generality. The idea of Bayesian updating of a prior with new information
seems to presuppose that the old and the new information refer to a common
relationship among the data
–
generality once more.
5 THE TRUTH IS SPECIALLY FITTED TO SURVIVE
The third attitude to data mining embraces the notion that there is a true data
generating process, although recognizing that we cannot ever be sure that we
have uncovered it. A good specificationsearch methodology is one in which
the truth is likely to emerge as the search continues on more and more data. On
this view, data mining is not a term of abuse but a description of an essential
empirical activity. The only issue is whether any particular data mining
scheme is a good one. This prodata mining attitude is most obvious in the so
called LSE (London School of Economics) methodology.
8
The relevant LSE
methodology is the generaltospecific modelling approach. It relies on an
intuitively appealing idea. A sufficiently complicated model can, in principle,
describe the economic world.
9
Any more parsimonious model is an improve
ment on such a complicated model if it conveys all of the same information in
a simpler, more compact form. Such a parsimonious model would necessarily
be superior to all other models that are restrictions of the completely general
model except, perhaps, to a class of models nested within the parsimonious
model itself. The art of model specification in the LSE framework is to seek
out models that are valid parsimonious restrictions of the completely general
model and, that are not redundant in the sense of having an even more parsi
monious model nested within them that also are valid restrictions of the com
pletely general model.
The generaltospecific modelling approach is related to the theory of
encompassing.
10
Roughly speaking, one model encompasses another if it con
veys all of the information conveyed by another model. It is easy to understand
the fundamental idea by considering two nonnested models of the same
dependent variable. Which is better? Consider a more general model that uses
the nonredundant union of the regressors of the two models. If model I
is a valid restriction of the more general model (e.g., based on an Ftest) and
model II is not, then model I encompasses model II. If model II is a valid
restriction and model I is not, then model II encompasses model I. In either
case, we know everything about the joint model from one of the restricted
models, we therefore know everything about the other restricted model from
that one. There is, of course, no necessity that either model will be a valid
Three attitudes towards data mining 203
restriction of the joint model: each could convey information that the other
failed to convey. A hierarchy of encompassing models arises naturally in a
generaltospecific modeling exercise. A model is tentatively admissible on
the LSE view if it is congruent with the data in the sense of being: (i) consistent
with the measuring system (e.g., not permitting negative fitted values in cases
in which the data are intrinsically positive); (ii) coherent with the data in that
its errors are innovations that are white noise as well as a martingale difference
sequence relative to the data considered; and (iii) stable (cf. Phillips 1988:
352
–
53; Mizon 1995: 115
–
22; White 1990: 370
–
74). Further conditions
(e.g., consistency with economic theory, weak exogeneity of the regressors
with respect to parameters of interest, orthogonality of decision variables)
may also be required for economic interpretability or, to support policy inter
ventions or other particular purposes. If a researcher begins with a tentatively
admissible general model and pursues a chain of simplifications, at each
step maintaining admissibility and checking whether the simplified model is a
valid restriction of the more general model, then the simplified model will be a
more parsimonious representation of all the models higher on that particular
chain of simplification and will encompass all of the models lower along the
same chain.
The generaltospecific approach might be seen as an example of invidious
data mining. The encompassing relationships that arise so naturally apply only
to a specific path of simplifications. One objection to the generaltospecific
approach is that there is no automatic encompassing relationship between
the final models of different researchers, who have wandered down different
paths in the forest of models nested in the general model. One answer to this
is that any two models can be tested for encompassing either through the
application of nonnested hypothesis tests or through the approach described
above, of nesting them within a joint model. Thus, the question of which, if
either, encompasses the other can be resolved, except in cases in which sample
size is inadequate.
A second objection notes that variables may be correlated either because
there is a genuine relation between them or because
–
in short samples
–
they
are adventitiously correlated. This is the objection of Hess et al. (1998) that
the generaltospecific specification search of Baba et al. (1992) selects
an ‘overfitting’ model. Any search algorithm that retains significant variables
will be subject to this objection since adventitious correlations are frequently
encountered in small samples. They can be eliminated only through an appeal
to wider criteria, such as agreement with a priori theory. One is entitled to ask
though, before accepting this criticism, on what basis should these criteria be
privileged?
By far the most common reaction of critical commentators and referees to
the generaltospecific approach questions the meaning of the test statistics
associated with the final model. The idea of Darwinian test statistics arises, as
it does for Lovell, because test statistics which are welldefined only under the
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correct specification, are compared across competing (and, therefore, necessarily
not all correct) specifications.
The generaltospecific approach is straightforward regarding this issue.
It accepts that choice among specifications is unavoidable, that an economic
interpretation requires correct specification and that correct specification
is not likely to be given a priori. The generaltospecific search treats and
focuses on the relationship between the specification and the data, rather than,
as is the case with the other two attitudes, on the relationship between the
investigator and the specification. That is, it interprets the test statistics as
evidence of sampling distribution rather than as measures of epistemic
warrant. Each specification is taken on probation. The question posed is
counterfactual: what would the sampling distributions be if the specification in
hand were in fact the truth? The true specification, for example, by virtue of
recapitulating the underlying datagenerating process, should show errors
that are white noise innovations. Similarly, the true specification should
encompass any other specification (in particular it should encompass the higher
dimensional general specification in which it is nested).
The generaltospecific approach is Darwinian but in a different sense than
that implied in the other two attitudes. The notion that only our prejudices
survive, or that the key issue is to modify critical values to account for the
degree of search, assumes that we should track some aspect say, the coefficient
on a particular variable, through a series of mutations (the alternative specifi
cations) and that the survival criterion is our particular prior commitment to
a value, sign or level of statistical significance for that variable. The general
tospecific methodology rejects the idea that it makes sense to track an aspect
of an evolving specification. Since the specification is regarded as informative
about the data rather than about the investigator or the history of the investi
gation, each specification must be evaluated independently. Nor should our
preconceptions serve as a survival index. Each specification is evaluated for
its verisimilitude (does it behave statistically like the truth would behave were
we to know the truth?) and, for its relative informativeness (does it encompass
alternative specifications?). The surviving specification in a search is a model
of the statistical properties of the data and identical specifications bear the
same relationship to the data whether that search was an arduous bit of data
mining or a directly intuited step to the final specification.
Should we expect the distillation process to lead to the truth? The Darwinian
nature of the generaltospecific search methodology can be explained with
reference to a remarkable theorem due to Halbert White (1990: 379
–
80). The
upshot of which is this: for a fixed set of specifications and a battery of specifi
cation tests, as the sample size grows toward infinity and increasingly smaller
test sizes are employed, the test battery will
–
with a probability approaching
unity
–
select the correct specification from the set. In such cases, White’s
theorem implies that type I and type II error both fall asymptotically to zero.
White’s theorem says that, given enough data, only the true specification will
Three attitudes towards data mining 205
survive a stringent enough set of tests. This turns the criticisms which regard
data mining as Darwinian, in a pejorative sense, on their heads. The critics fear
that the survivor of sequential tests survives accidentally and, therefore, that
the critical values of such tests ought to be adjusted to reflect the likelihood
of an accident. White’s theorem suggests that the true specification survives
precisely because the true specification is necessarily, in the long run, the
fittest specification.
Approaches that focus on correcting critical values miss the point. White’s
theorem suggests that we envisage the problem differently. An analogy is the
fitting together of a jigsaw puzzle. Even if a piece duplicates another in part or
all of its shape, as more pieces are put into place, the requirement that the
surface picture as well as the geometry of the pieces cohere implies that each
piece has a unique position. Inferences about the puzzle as a whole, or
the piece in relation to the puzzle, can be made soundly only conditional on
getting the pieces into their proper positions. And, in the long run, the puzzle
fits together only one way
–
a fact about the puzzle itself, not about us.
White’s theorem is an asymptotic result. The real world of economics does
not deal in infinite samples of data. But the generaltospecific methodology
proceeds from a similar vision of the relationship of testing to the truth. The
interesting methodological question on this view is, what are effective pro
cedures for solving the jigsaw puzzles of economics when samples are small?
We have made a first pass at this question.
Hoover and Perez (1999) evaluate the generaltospecific methodology in a
simulation study inspired by Lovell’s (1983) Monte Carlo study of mechan
ized search algorithms (see section 3 above). Lovell concludes that the three
simple algorithms he examines (stepwise regression, maximum R
2
and max
min tstatistics) are all quite poor in recovering the true datagenerating
processes. Furthermore, the size and power of the algorithms taken as a whole
(that is the ability of the algorithms to exclude variables that were not in
the datagenerating process and to include variables that were) is rather poor
and quite different from that implied by conventional critical values based on
sampling distributions. Using updated data and the same datagenerating pro
cesses, we are able to confirm Lovell’s results for the algorithms he tests on
annual data.
We then extend the investigation to the generaltospecific search proce
dure. We use the same variables but at a quarterly frequency and we difference
each series until it is stationary on standard tests. In the hands of econo
metricians of the LSE school, the generaltospecific methodology is not
mechanical like Lovell’s stepwise regression or other algorithms. Neverthe
less, we have developed a mechanical algorithm that mimics some key features
of the generaltospecific approach. It begins with a general model in which
the entire set of variables in Lovell’s data set (included lagged variables) appears
as regressors. This regression is tested for congruence. If it passes, simplifi
cation begins. Regressors with the low tstatistics are deleted in sequence,
206 Articles
starting a different search with each of the ten lowest, providing that they
are insignificant at conventional sizes. When a regressor is deleted and a new
estimate obtained, it is checked to see whether it encompasses the general
model. If it does not or if it fails any tests of congruence, the regressor is
replaced and the variable with the next lowest tstatistic from the previous
regression is eliminated instead. If it does, then the variable with the next lowest
insignificant tstatistic in the current regression is eliminated and, the process
of testing is repeated. Elimination continues until either all retained variables
are significant, no variables can be removed without failing congruence, or no
variables can be removed without failing to encompass the general model. Ten
such regressions from ten search paths, corresponding to the ten regressors
with lowest tstatistics in the general model, constitute the possible character
izations of the data. The model among these ten that encompasses the others is
chosen as the final model.
11
Lovell, and proponents of the first view of datamining would have us
report this specification but, adjust the standard errors of the coefficients to
account for the many regressions run. Leamer and proponents of the second
view of data mining would have us look at all of the many regressions and
choose only those variables that do not have their coefficient values change
significantly over the search, i.e. the end result is not important, only the distri
bution of the coefficients estimates is important. The generaltospecific approach
takes the final specification to represent the best approximation to the truth
because it acts as closely to how the truth would act, if we knew it. Evolution is
important but not the history of evolution.
To check the success of this algorithm, we simulate with the updated quarterly
data the same models that Lovell uses. There are nine models, static and
dynamic, calibrated from the actual data. The independent variables are actual
data but the dependent variable is simulated from the actual independent
variables and draws from a random distribution with characteristics that match
the performance of each model in actual data. We conduct 1000 simulations
and searches of each model. In contrast to Lovell’s algorithms, we find that
the generaltospecific methodology is effective at recovering the true data
generating process. It does not always succeed. Where it fails, it seems to be
almost exclusively because of low signaltonoise ratios. This is reflected in
the fact that both size and power measures are close to what one obtains from
Monte Carlo simulations in which the true specification is known in advance.
It also shows that no method could be expected in some cases to find the true
model when there is simply not enough information in the data. Our results are
supportive of the generaltospecific methodology but they are limited. Work
inprogress aims to extend the evaluation to nonstationary data, in which
questions of cointegration are important, and to crosssectional contexts.
The point of this long digression is not principally to advertise our own
work although we are happy if it does that. Rather it is to demonstrate an
empirical spirit. We hope to have provided a theoretical analysis of why the
Three attitudes towards data mining 207
concerns of the opponents of data mining are misplaced but we also wish
to allay any nagging doubts by pointing to evidence that, practically, the
difficulties they foresee do not in fact arise.
6 CONCLUSION
Econometrics is a tool for learning about the economy. It presupposes the
existence of an economy to learn about and not an assortment of facts but, an
economy with real general features that imply the behaviour of the data and
constrain the way that econometric and statistical calculations package the
data. There is a truth about the economy that remains the truth, even though it
presents a different aspect when viewed through different econometric filters,
just as there is a truth about the moon or the crab nebula that remains the truth
even though these heavenly bodies appear differently when viewed through
different telescopes and optical filters. The central issue is how to use these
observations most effectively. The school of thought that argues that the more
the search, the lower the epistemic warrant would reject a detailed picture
of a distant galaxy because it was difficult to obtain. It concentrates on the
astronomer, rather than on the astronomical object. The problem is that
the object may be objectively difficult to find and may yield only to highly
structured search. Indeed, many scientific results are valued precisely because
they are difficult to obtain. The school of thought that seeks robustness rather
than truth would reject the picture because many other pictures of the same
sector of the sky processed in different ways look quite different. Whereas the
school of thought that we endorse argues that the issues that need to be
addressed are, first, whether the lenses and filters of the astronomers or the
statistics of the econometricians are in fact the ones that would reveal the
aspect of the truth that interests us
–
if it were there
–
and, second, whether, in
the event, they do so. A regulated specification search, such as the generalto
specific methodology proposes, is an attempt to use econometrics to bring an
economic reality into focus that would otherwise remain hidden. It aims, quite
literally, to discover the truth.
Kevin D. Hoover
Department of Economics, University of California, USA
kdhoover@ucdavis.edu
Stephen J. Perez
Department of Economics, Washington State University, USA
sjperez@wsu.edu
ACKNOWLEDGEMENTS
The authors wish to thank Roger Backhouse, Peter Burridge, Alistair Hall,
Thomas Mayer, Mary Morgan and Steven Sheffrin for comments on earlier
drafts.
208 Articles
NOTES
1 This is not to deny that analytical conclusions are drawn with respect to models
with stochastic regressors. But even such models are characterized at some
level by probability distributions with constant parameters (cf. the debate
between Leamer and Hendry in Hendry et al. (1990: 195).
2 To keep the discussion simple, we will often speak of the standard errors and
tstatistics of regression coefficients as exemplars of the sampling distribution.
Our points are also relevant mutatis mutandis to other statistics.
3 On Monte Carlo methods, see Hendry (1995, ch. 3, section 6); on bootstrap
methods, see Jeong and Maddala (1993).
4 Sampling distribution and epistemic warrant are two distinct things. In particular,
epistemic warrant refers to the circumstances in which we have adequate justi
fication for a belief. It is not the perfection of the sampling distribution in the
sense of being the population distribution that corresponds to the sample.
5 These categories are drawn from McAleer et al. (1985) and their restatement of
Leamer’s extreme bounds analysis.
6 It is important not to confuse the idea of robustness here with notions of temporal
stability or homogeneity among subsamples. Truth need not be constant over time
nor global for a given set of data. Robustness to alternative specification holds the
sample constant, so these questions do not arise; it is only the specification that
varies.
7 The point estimate itself will not switch signs if the regressors are orthogonal as
supposed but, might do so if they are correlated with each other.
8 The LSE approach is described sympathetically in Gilbert (1986), Hendry (1987,
1995, especially ch. 9–15), Pagan (1987), Phillips (1988), Ericsson et al. (1990)
and Mizon (1995). For more sceptical accounts, see Faust and Whiteman (1995)
and Hansen (1996). The adjective ‘LSE’ is, to some extent, a misnomer. It derives
from the fact that there is a tradition of timeseries econometrics that began in the
1960s at the London School of Economics, see Mizon (1995) for a brief history.
The practitioners of LSE econometrics are now widely dispersed among aca
demic institutions throughout Britain and the world.
9 This is a truism. Practically, however, it involves a leap of faith; for models that
are onetoone, or even distantly approach onetoone, with the world are not
tractable.
10 For general discussions of encompassing, see, for example, Mizon (1984), Mizon
and Richard (1986), Hendry and Richard (1987), Hendry (1988, 1995 ch. 14).
11 In this paper we use the necessary condition for encompassing that the standard
error of regression for the encompassing model must be the lowest of the ten. In
work in progress we conduct a sequence of encompassing tests to choose among
the ten, allowing for the fact that some linear combination of them might in fact
encompass the others, even while none of them does so individually.
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